6D F-theory models and elliptically fibered Calabi-Yau
threefolds over semi-toric base surfaces
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Citation
Martini, Gabriella, and Washington Taylor. “6D F-Theory Models
and Elliptically Fibered Calabi-Yau Threefolds over Semi-Toric
Base Surfaces.” J. High Energ. Phys. 2015, no. 6 (June 2015).
As Published
http://dx.doi.org/10.1007/JHEP06(2015)061
Publisher
Springer-Verlag
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Final published version
Accessed
Fri May 27 05:27:26 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/98246
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http://creativecommons.org/licenses/by/4.0/
Published for SISSA by
Springer
Received: January 27, 2015
Accepted: May 15, 2015
Published: June 10, 2015
Gabriella Martini and Washington Taylor
Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
E-mail: gmartini@mit.edu, wati@mit.edu
Abstract: We carry out a systematic study of a class of 6D F-theory models and associated Calabi-Yau threefolds that are constructed using base surfaces with a generalization of
toric structure. In particular, we determine all smooth surfaces with a structure invariant
under a single C∗ action (sometimes called “T-varieties” in the mathematical literature)
that can act as bases for an elliptic fibration with section of a Calabi-Yau threefold. We
identify 162,404 distinct bases, which include as a subset the previously studied set of
strictly toric bases. Calabi-Yau threefolds constructed in this fashion include examples
with previously unknown Hodge numbers. There are also bases over which the generic
elliptic fibration has a Mordell-Weil group of sections with nonzero rank, corresponding to
non-Higgsable U(1) factors in the 6D supergravity model; this type of structure does not
arise for generic elliptic fibrations in the purely toric context.
Keywords: F-Theory, Differential and Algebraic Geometry, Supergravity Models
ArXiv ePrint: 1404.6300
c The Authors.
Open Access, Article funded by SCOAP3 .
doi:10.1007/JHEP06(2015)061
JHEP06(2015)061
6D F-theory models and elliptically fibered Calabi-Yau
threefolds over semi-toric base surfaces
Contents
1
2 C∗ -surfaces and F-theory vacua
2.1 6D F-theory models
2.2 General classification of 6D F-theory base surfaces
2.3 Toric base surfaces
2.4 C∗ -base surfaces
2.5 Curves of self-intersection -9, -10, and -11
3
3
3
5
7
8
3 Enumeration of bases
3.1 Classification and enumeration of C∗ -bases
3.2 Distribution of bases
3.3 Gauge groups and chain structure
9
9
10
14
4 Calabi-Yau geometry
4.1 Calabi-Yau geometry of elliptic fibrations over C∗ -bases
4.2 Counting neutral hypermultiplets
4.3 Distribution of Hodge numbers
4.4 Redundancies from −2 clusters
4.5 Calabi-Yau threefolds with new Hodge numbers
4.6 Calabi-Yau threefolds with nontrivial Mordell-Weil rank
16
17
18
22
23
26
26
5 Conclusions
28
A Rules for connecting clusters in C∗ -bases
30
B Bases giving Calabi-Yau threefolds with new Hodge numbers
32
C Bases giving U(1) factors
33
1
Introduction
Since the early days of string theory, much effort has been devoted to understanding the
geometry of string compactifications. Calabi-Yau threefolds are one of the most central
and best-studied classes of compactification geometries. These manifolds can be used to
compactify ten-dimensional superstring theories to give four-dimensional supersymmetric
theories of gravity and gauge fields [1, 2]. Calabi-Yau threefolds that admit an elliptic
fibration with section can also be used to compactify F-theory to give six-dimensional
supergravity theories [3–5].
–1–
JHEP06(2015)061
1 Introduction
1
Throughout this paper we use the term “base surface” as shorthand for “surface that can act as the
base of an elliptically-fibered Calabi-Yau threefold with section”.
–2–
JHEP06(2015)061
While mathematicians and physicists have used many methods to construct and study
Calabi-Yau threefolds (see [6] for a recent review), one of the main approaches that has
been fruitful for systematically classifying large numbers of Calabi-Yau geometries is the
mathematical framework of toric geometry [7]. The power of toric geometry is that many
geometric features of a space are captured in a simple combinatorial framework that lends
itself to straightforward calculations for many quantities of interest. An approach was
developed by Batyrev [8] for describing Calabi-Yau manifolds as hypersurfaces in toric
varieties in terms of the combinatorics of reflexive polytopes. Kreuzer and Skarke have
identified some 473.8 million four-dimensional reflexive polytopes that can be used to construct Calabi-Yau threefolds in this way [9].
In this paper, following [10], we study a class of spaces that is more general than the set
of toric varieties, but retains some of the combinatorial simplicity of toric geometry. This
allows us to construct a large class of elliptically fibered Calabi-Yau threefolds that need
not have any realization in a strictly toric language. In particular, we focus on complex
surfaces that admit at least one C∗ action but not necessarily the action of a product C∗ ×C∗
needed for a full toric structure. Complex algebraic varieties of dimension n that admit
an effective action of (C∗ )k are known in the mathematical literature as “T-varieties”. In
this paper we simply use the term “C∗ -surface” to denote a surface with a C∗ action, in
part to avoid confusion with the plethora of “T-” objects already filling the string theory
literature (“T-duality”, “T-folds”, “T-branes”, etc.) A review of some of the mathematical
results and literature on more general T-varieties can be found in [11].
The primary focus of this paper is the systematic classification and study of all smooth
C∗ -surfaces that can act as bases for an elliptic fibration with section where the total space
is a Calabi-Yau threefold. We use these surfaces to compactify F-theory to six dimensions.
The close correspondence between the geometry of the F-theory compactification surface
and the physics of the associated 6D supergravity theory provides a powerful tool for
understanding both geometry and physics. A general characterization of smooth base
surfaces1 (not necessarily toric or C∗ -) was given in [10]. The basic idea is that any
base can be classified by the intersection structure of effective irreducible divisors of selfintersection −2 or below. The intersection structure of the base directly corresponds to the
generic nonabelian gauge group in the “maximally Higgsed” 6D supergravity theory from
an F-theory construction. There are strong constraints on the intersection structures that
can arise; these constraints made possible an explicit enumeration of all toric base surfaces
in [12], and the geometry of the associated Calabi-Yau threefolds was explored in [13]. In
this paper we construct and enumerate the more general class of C∗ -bases and explore the
corresponding Calabi-Yau geometries. In particular, we identify some (apparently) new
Calabi-Yau threefolds including some threefolds with novel properties.
In section 2 we describe the class of C∗ -base surfaces. We summarize the results of the
complete enumeration of these bases in section 3. The corresponding Calabi-Yau threefolds,
including some models with interesting new features, are described in section 4. Section 5
contains concluding remarks. Appendices A–C contain tables of useful data.
2
C∗ -surfaces and F-theory vacua
We begin by briefly summarizing the results of [10, 12]. The methods used here are closely
related to those developed in these papers.
2.1
6D F-theory models
where f, g are sections of the line bundles O(−4K), O(−6K), with K the canonical class
of B. Codimension one vanishing loci of f, g, and the discriminant locus ∆ = 4f 3 + 27g 2 ,
where the elliptic fibration becomes singular, give rise to vector multiplets for a nonabelian
gauge group in the 6D theory. Codimension two vanishing loci give matter in the 6D theory;
the matter lives in a representation of the nonabelian gauge group that is determined by
the geometry. For more detailed background regarding the relation of 6D physics to Ftheory geometry, see the reviews [14–16]; a recent systematic analysis of these 6D models
from the M-theory point of view appears in [17]).
2.2
General classification of 6D F-theory base surfaces
A general approach to systematically classifying base surfaces B for 6D F-theory compactifications was developed in [10]. This approach is based on identifying irreducible
components in the structure of effective divisors on B, composed of intersecting combinations of curves of negative self-intersection over which the generic elliptic fibration is
singular. Each such irreducible component corresponds to a “non-Higgsable cluster” of
gauge algebra summands and (in some cases) charged matter that appears in the 6D supergravity model arising from an F-theory compactification on a generic elliptic fibration
over B; the term “non-Higgsable cluster” (which we sometimes abbreviate as “cluster”)
refers to the fact that for any such configuration there are no matter fields that can lead to
a Higgsing of the corresponding nonabelian gauge group. For example, a single irreducible
effective curve in B with self-intersection −4 corresponds to an so(8) term in the gauge
algebra with no charged matter. The simplest example of this occurs for F-theory on the
Hirzebruch surface F4 [4, 5]. There are strong geometric constraints (discussed further below) on the configurations of curves that can live in a base supporting an elliptic fibration.
The set of possible non-Higgsable clusters is thus rather small. A complete list is given in
table 1, and depicted in figure 1. These clusters are described by configurations of curves
having self-intersection −2 or less, with one curve having self-intersection −3 or below.
General configurations of −2 curves with no curves of self-intersection −3 or below are also
possible. These clusters carry no gauge group, and generally represent limiting points of
–3–
JHEP06(2015)061
A 6D F-theory model is defined by a Calabi-Yau threefold that is an elliptic fibration (with
section) over a complex base surface B. The structure of the resulting 6D supergravity
theory is determined by the geometry of B and the elliptic fibration [4, 5]. In particular,
the number of tensor multiplets T in the 6D theory is related to the topology of the base
B through T = h1,1 (B) − 1. The elliptic fibration can be described by a Weierstrass model
over B
y 2 = x3 + f x + g ,
(2.1)
gauge algebra
e8
e7
e7
e6
f4
so(8)
g2 ⊕ su(2)
g2 ⊕ su(2)
su(3)
su(2) ⊕ so(7) ⊕ su(2)
no gauge group
Hcharged
0
0
28
0
0
0
8
8
0
16
0
Table 1. Allowed “non-Higgsable clusters” of irreducible effective divisors with self-intersection
−2 or below, and corresponding contributions to the gauge algebra and matter content of the 6D
theory associated with F-theory compactifications on a generic elliptic fibration (with section) over
a base containing each cluster.
−m ∈
{−3, −4, . . . , −8, −12}
@ -3
@
@
@ -3
@
@
@
-2
@
@
-2
su(3), so(8), f4
e6 , e7 , e8
@ -2
@
@
@
@
@ -2
@
@
g2 ⊕ su(2)
-3
g2 ⊕ su(2)
@ -2
@
@
su(2) ⊕ so(7) ⊕ su(2)
Figure 1. Clusters of intersecting curves that must carry a nonabelian gauge group factor. For
each cluster the corresponding gauge algebra is noted and the gauge algebra and number of charged
matter hypermultiplet are listed in table 1.
bases without these −2 curves; for example, the Hirzebruch surface F2 contains a single
isolated −2 curve, and is a limit of the surface F0 .
The restriction on the types of allowed clusters comes from the constraint that f, g
cannot vanish to degrees 4, 6 respectively on any curve or at any point in B. If the degree
of vanishing is too high on a curve, there is no way to construct a Calabi-Yau from the
elliptic fibration. If the vanishing is too high at a point, the point must be blown up to form
another base B 0 that supports a Calabi-Yau. The vanishing degrees of f, g on a curve can
be determined from the Zariski decomposition of −nK: using the fact that any effective
divisor A that has a negative intersection with an effective irreducible B having negative
self-intersection must contain B as a component (A · B < 0, B · B < 0 ⇒ A = B + C with
C effective), the degree of vanishing of A = −4K, −6K, −12K on any irreducible effective
–4–
JHEP06(2015)061
Cluster
(-12)
(-8)
(-7)
(-6)
(-5)
(-4)
(-3, -2, -2)
(-3, -2)
(-3)
(-2, -3, -2)
(-2, -2, . . . , -2)
2.3
Toric base surfaces
The set of toric base surfaces form a subset of the more general class of C∗ -bases that we
focus on in this paper. The structure of toric bases is relatively simple and provides a
useful foundation for our analysis of C∗ -bases.
2
aside from the Enriques surface, which gives rise to a simple 6D model with no gauge group or matter
and is connected in a more complicated way to the branches associated with other bases.
–5–
JHEP06(2015)061
curve or combination of curves can be computed; the degree of vanishing at a point where
two curves intersect is simply the sum of the degrees of vanishing on the curves. The details
of this computation for general combinations of intersecting divisors are worked out in [10].
All effective irreducible curves of self-intersection −2 or below in the base appear in
the clusters described above. Furthermore, again because of restrictions on the degrees of
f, g on curves and intersection points, only certain combinations of the allowed clusters
can be connected by −1 curves in B. For example, if a −1 curve intersects a −12 curve,
then the −1 curve cannot intersect any other curve contained in a non-Higgsable cluster
except a −2 curve contained in a (−2, −2, −3) cluster. A complete table of the possible
combinations of clusters that can be connected by a −1 curve is given in [10]. The part of
that table that is relevant for this paper is reproduced in appendix A.
For any base surface B, the structure of non-Higgsable clusters determines the minimal
nonabelian gauge group and matter content of the 6D supergravity theory corresponding
to a generic elliptic fibration with section over B. For each distinct base B there can be
a wide range of models with different nonabelian gauge groups, which can be realized by
tuning the Weierstrass model to increase the degrees of f, g over various divisors. For example, for the simplest base surface B = P2 , there are thousands of branches of the theory
with different nonabelian gauge group and matter content, some of which are explored
in [18, 19]. Each of these branches corresponds to a different Calabi-Yau threefold after
the singularities associated with the nonabelian gauge group factors are resolved. But for
each of these models, by maximally Higgsing matter fields in the supergravity theory the
gauge group can be completely broken and the geometry becomes that of a generic elliptic
fibration over P2 . Focusing on the base surface B dramatically simplifies the problem of
classifying F-theory compactifications and elliptically fibered Calabi-Yau threefolds, by removing the additional complexity associated with the details of specific Weierstrass models
and associated fibrations.
The mathematics of minimal surface theory [20, 21] gives a simple picture of how the
set of allowed base surfaces B are connected, unifying the space of 6D supergravity theories
that arise from F-theory. All smooth bases2 B for 6D F-theory models can be constructed
by blowing up a finite set of points on one of the minimal bases Fm (0 ≤ m ≤ 12, m 6= 1)
or P2 [22]. The number of distinct topological types for B is finite [23, 24]. In principle,
all smooth bases B can be systematically constructed by successively blowing up points on
the minimal bases allowing only the clusters of irreducible divisors from table 1. In [12],
the complete set of toric bases was constructed in this fashion, and in this paper we carry
out the analogous construction for C∗ -bases.
C1 (D0 )
+2
C4 0
0 C2
−2
C3 (D∞ )
+1
−1 A
A
−1 A
A
0
−1 0
- −2
0
\
−1\\
−2
−2
D0
−1 0 A −1
A
- −2
AA
\
\ −3
−1
−1 \
D∞
A toric surface can be described alternatively in terms of the toric fan or in terms of the
set of toric effective divisors. The fan for a compact toric surface is defined [7] by a sequence
of vectors v1 , . . . , vk ∈ N = Z2 defining 1D cones, or rays, along with 2D cones spanned
by vectors vi , vi+1 (including a 2D cone spanned by vk , v1 ; all related conditions include
this periodicity though we do not repeat this explicitly in each case), and the 0D cone at
the origin. The origin represents the torus (C∗ )2 , while the 1D rays represent divisors and
the 2D cones represent points in the compact toric surface. The surface is smooth if the
rays vi , vi+1 defining each 2D cone span a unit cell in the lattice. The irreducible effective
toric divisors are a set of curves3 Ci , i = 1, . . . , k, associated with the vectors vi . The
divisors have self-intersection Ci · Ci = ni , where −nvi = vi−1 + vi+1 , and nonvanishing
pairwise intersections Ci · Ci+1 = Ck · C1 = 1. These divisors can be depicted graphically
as a loop (See figure 2). The sets of consecutive divisors of self-intersection ni ≤ −2
in the loop are constrained by the cluster analysis of [10] to contain only the sequences
(with either orientation) (−3, −2), (−3, −2, −2), (−2, −3, −2), (−m), with 3 ≤ m ≤ 12,
and (−2, . . . , −2) with any number of −2 curves.
All smooth toric base surfaces (aside from P2 ) can be constructed by starting with
a Hirzebruch surface Fm that is associated with the divisor self-intersection sequence
[n1 , n2 , n3 , n4 ] = [m, 0, −m, 0] (v1 = (0, 1), v2 = (1, 0), v3 = (0, −1), v4 = (−1, −m)), and
blowing up a sequence of intersection points between adjacent divisors. Blowing up the
intersection point between divisors Ci and Ci+1 gives a new toric base with a -1 curve inserted between these divisors, and self-intersections of the previously intersecting divisors
each reduced by one (See figure 2). Such blow-ups are the only ones possible that maintain
the toric structure by preserving the action of (C∗ )2 on the base. Each Hirzebruch surface
can be viewed as a P1 bundle over P1 . The divisors of self-intersection ±m can be viewed as
sections Σ± of this bundle, which in a local coordinate chart (z, w) ∈ C×C are at the points
(z, 0), (z, ∞) in the fibers, while the divisors of self-intersection 0 can be viewed as fibers
(0, w) and (∞, w). All points that can be blown up while maintaining the toric structure
are located at the points invariant under the (C∗ )2 action: (0, 0), (0, ∞), (∞, 0), (∞, ∞)
(or in exceptional divisors produced when these points are blown up). In particular, any
3
Note that the notation and ordering used here for curves and associated rays in a toric base differs from
that used in [12].
–6–
JHEP06(2015)061
Figure 2. Toric base surfaces for 6D F-theory models produced by blowing up a sequence of
points on F2 .
2.4
C∗ -base surfaces
A more general class of bases B was described in [12]. Generalizing the toric construction
by allowing blow-ups at arbitrary points (z, 0), (z, ∞) can give rise to an arbitrary number
of fibers containing curves of negative self-intersection, while maintaining the C∗ action
(1 × C∗ ) that leaves the sections Σ± invariant. The resulting structure is closely analogous
to that of the toric bases described above, except that there can be more than two chains
of intersecting divisors connecting D0 , D∞ , associated with distinct blown-up fibers of the
original Hirzebruch surface. Graphically, the intersection structure of effective irreducible
divisors for such a base can be depicted as the pair of horizontal divisors D0 , D∞ (associated
with Σ± in the original Hirzebruch surface), along with an arbitrary number N of chains
of divisors Di,j , i = 1, . . . , N connecting D0 , D∞ . The ith chain is realized by blowing up
a fiber in Fm , and contains divisors Di,j , j = 1, . . . , ki , with self-intersections and adjacent
intersections as described above (Di,j · Di,j = ni,j < 0, Di,j · Di,j+1 = 1) except that
Di,1 · D0 = 1, Di,ki · D∞ = 1, and Di,1 · Di,ki = 0 unless ki = 2. (See figure 3.) We refer
to bases of this form as C∗ -bases. In this language, the toric bases correspond to those
C∗ -bases with N = 2 (or fewer) chains. The divisor intersection structure of a C∗ -base
is constrained by the set of allowed clusters found in [10] in a similar way to toric bases,
but with additional possibilities associated with the existence of multiple connections to
the divisors D0 , D∞ that provide branchings in the intersection structure. For example, if
D0 has self-intersection D0 · D0 = −3, and is connected to chains with self-intersections
(n1,1 , n1,2 , . . .) = (−2, −1, . . .), (n2,1 , n2,2 , . . .) = (−2, −1, . . .), then any further chains i > 2
must satisfy ni,1 = −1, ni,2 ≥ −4. The complete set of constraints on how divisors can be
connected is described in appendix A, including some additional constraints beyond those
described in [10] related to branchings on curves of self-intersection −n < −1.
Note that the value of T = h1,1 (B)−1 can be determined directly from the intersection
structure of a C∗ -base. Each blow-up adds one to T . Each blow-up along a new fiber
increases N by one and creates a new (−1, −1) chain with kN = 2, while each blow-up on
an intersection in chain i increases ki by 1. Matching with the Hirzebruch surfaces, which
have N = 0, T = 1, we have
!
N
X
1,1
T = h (B) − 1 =
ki − N + 1 .
(2.2)
i=1
–7–
JHEP06(2015)061
smooth toric base surface that supports an elliptically fibered Calabi-Yau threefold has
a description in terms of a closed loop of divisors containing divisors D0 , D∞ associated
with the divisors C1 , C3 in the original Hirzebruch surface (these are the sections Σ± ) and
two linear chains of divisors connecting D0 to D∞ . These chains are formed from the
divisors C2 and C4 in the original Hirzebruch surface, along with all exceptional divisors
from blowing up points on these original curves. This is illustrated in figure 2.
Note that some smooth toric base surfaces can be formed in different ways from blowups of Hirzebruch surfaces, so that different divisors D0 , D∞ play the role of the sections Σ± .
In enumerating all toric base surfaces, such duplicates must be eliminated by considering
equivalences of the loops of self-intersection numbers up to rotation and reflection.
D0
-3
D1,1 -2
D2,1 -1
J
D3,1J -1
J
J
D1,2J -1
J
J
J
J
J
D2,2J -2
J
J
J
D3,2
J
D2,3
-2
-2
J
J
D2,4J -1
J
J
-4
D3,3 -1
D∞
Figure 3. A C∗ -base B is characterized by irreducible effective divisors D0 , D∞ connected by any
number of linear chains of divisors Di,j with intersections obeying the cluster rules of [10]. All such
bases can be realized as multiple blow-ups of a Hirzebruch surface Fm that preserve the action of
a single C∗ on B.
2.5
Curves of self-intersection -9, -10, and -11
There is one further issue that arises in systematically classifying toric and/or C∗ -bases. As
shown in [10], for any base containing a curve C of self-intersection −9, −10, or −11, there
is a point on C where the Weierstrass functions f, g vanish to degrees 4, 6, so that point
in the base must be blown up for the base to support an elliptically fibered Calabi-Yau
threefold. If the curve C is a divisor associated with a section, D0 or D∞ , then blowing
up this point simply adds another (−1, −1) chain to the C∗ -surface (figure 4). If C is an
element of one of the chains, however, then blowing up the point on C generally takes the
base out of the class of C∗ - or toric surfaces.
For strictly toric or C∗ -surfaces, therefore, we should not include any bases containing
curves of self-intersection -9, -10, or -11. For several reasons, however, we find it of interest
to include bases with such curves even when the blow-up leaves the context of C∗ -surfaces.
In particular, we are interested in exploring the widest range of bases possible that can
be systematically analyzed. Thus, while the bases arising from blowing up -9, -10, or 11 curves on connecting chains generally takes us outside the C∗ context we have done
a complete enumeration of surfaces including these additional types of curves (which we
refer to as “not-strictly C∗ -bases,” or “NSC-bases” for short), with the understanding that
the blown-up smooth surfaces that support elliptic Calabi-Yau fibrations are no longer
strictly C∗ -surfaces. One strong argument in favor of including these bases in our analysis
is that the base with the largest known value of h1,1 (B) = 491, corresponding to the 6D
F-theory compactification with the largest gauge group, is of this type. This geometry was
first identified in [25, 26], and was studied further in [12, 13]. This geometry arises from
a toric base that contains two −11 curves that cannot be associated with D0,∞ , so the
–8–
JHEP06(2015)061
D1,3 -3
J
J
D1,4J -1
J
J
D0
0
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
B B B B B B B B B B B 0
-1B -1B -1B -1B -1B -1B -1B -1B -1B -1B -1B
B B B B B B B B B B B
-11
D0
0
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-
B
B
B
B
B
B
B
B
B
B
B
B
-1B -1B -1B -1B -1B -1B -1B -1B -1B -1B -1B -1B
B
D∞
B
B
B
B
B
-12
B
B
B
B
B
B
D∞
blown-up smooth base surface is neither strictly toric nor C∗ . We included toric bases with
−9, −10, −11 curves in [12]. In the next section we discuss how these are systematically
included in the enumeration of C∗ -bases in the analysis of this paper.
For (NSC) surfaces that contain -9, -10, or -11 curves, an additional contribution to
T must be added for each blow-up needed to reach a smooth surface with only allowed
clusters (i.e., with -12 curves instead), so the equation for T is modified from (2.2) to
!
N
X
1,1
T = h (B) − 1 =
ki − N + c11 + 2c10 + 3c9 + 1 ,
(2.3)
i=1
where ck is the number of curves with self-intersection −k.
3
Enumeration of bases
We now summarize the results of a full enumeration of the set of C∗ -base surfaces relevant
for F-theory compactification. In this section we describe the features relevant for the
corresponding 6D supergravity theories. The following section (section 4) focuses on aspects
of the associated Calabi-Yau geometries.
3.1
Classification and enumeration of C∗ -bases
Any C∗ -surface can be characterized by the self-intersection numbers n0 , n∞ of the divisors
D0 , D∞ associated with the sections Σ± , the number N of chains associated with distinct
fibers along which blow-ups have occurred, and the intersection structure of the divisors
connected along each chain, characterized by the integers ni,j as described above. Thinking
of each base B as a blow-up of Fm , each chain is constructed by first blowing up a point
at the intersection of a fiber Fi in Fm with either D0 or D∞ , giving a chain of length 2
containing a pair of intersecting −1 curves. Additional blow-ups can then be performed at
the intersections either between pairs of adjacent divisors along the chain, or between the
divisors at the end of the chain and D0 or D∞ . Each nontrivial chain thus is associated
–9–
JHEP06(2015)061
Figure 4. The C∗ -base with N = 11, n0 = 0, n∞ = −11 and 11 (−1, −1) chains has a singular
point on D∞ where the base must be blown up, giving the smooth base with N = 12, n∞ = −12,
and 12 (−1, −1) chains.
3.2
Distribution of bases
The number of distinct bases, uniquely determined by N, n0 , n∞ , and the intersection
configuration of the curves on the N chains (up to the symmetries associated with permutations of the chains and the simultaneous reflection of all chains combined with n0 ↔ n∞ ),
is tabulated for each N from 0 to 24 in table 2. Note that in cases where the base is toric
and could have either N = 1 or N = 2 (this can occur when there is a curve of selfintersection 0 that can either appear as a fiber or as one of D0 , D∞ ) we have counted the
base as having the minimal value, N = 1. We find a total of 126,469 smooth C∗ -bases that
are acceptable for 6D F-theory compactifications. There are an additional 35,935 bases
that come from C∗ -bases with -9, -10, and -11 curves in the fiber chains, for a total of
162,404 allowed bases. We use this larger set for the statistical analyses in the remainder
of this paper. The largest fraction (nearly 38%) of these bases come from the case with 3
– 10 –
JHEP06(2015)061
with a decrease in the self-intersection of D0 or D∞ by at least one. Since the minimum
values of n0 , n∞ are −12 (as determined by the allowed non-Higgsable clusters), and their
associated intersection numbers for Σ± on Fm are equal and opposite, the maximum number
of possible nontrivial chains is N = 24. Constructions with N = 2 or fewer nontrivial chains
are realized already in the toric context described in [12]; we have included these cases in
the complete analysis here, though the counting is slightly different due to the treatment
of −9, −10, and −11 curves.
To enumerate all possible C∗ -bases then, we can proceed by first constructing all
possible nontrivial chains associated with at most 24 blow-ups of points on the ending
divisors of each chain (similar constructions were described in [12, 27]). We then consider all
possible combinations of these chains compatible with the bound on intersection numbers
of the sections D0,∞ . To implement this enumeration algorithmically, we consider the set
of partitions of k such that 1 ≤ k ≤ 24, which are equivalent to all valid combinations
for the number of blow-ups at the intersection of distinct fibers with D0 or D∞ for any
Fm . Then for each Fm and partition λ ` k such that k = k1 + . . . + kN , we can identify
all combinations of chains associated with k1 , . . . , kN blow-ups on the ending divisors such
that the intersection of these chains with D0 and D∞ collectively satisfy the F-theory rules
contained in table 1 and table 3. Doing so for every achievable combination of n0 , n∞ with
λ ` k gives a systematic method for enumerating all valid C∗ -bases.
We have carried out the complete enumeration of all strictly C∗ -bases (i.e., including
no -9, -10, -11 curves) for all values of N from 0 to 24. We have also enumerated those
bases which are not strictly C∗ (or toric), due to -9, -10, or -11 curves on the fiber chains.
We have not considered bases with -9, -10, or -11 curves for D0 or D∞ , since blowing up
these curves gives a C∗ -base with larger N , so including these would simply amount to
overcounting. The cases N = 0, 1, 2 represent toric bases. We have not included toric
bases with -9, -10, -11 curves on fiber chains when there is an equivalent toric base where
such a curve can be mapped to D0 or D∞ by a rotation of the loop of toric divisors. These
cases are already counted elsewhere in the set of C∗ bases, as discussed in more detail in
the following section.
C∗ -bases
1
10
9,383
25,474
44,930
20,980
11,027
6,137
3,485
2,034
1,190
709
423
262
159
101
62
40
24
16
9
6
3
2
1
1
126,469
C∗ + NSC bases
1
10
14,183
28,733
61,329
27,134
13,811
7,462
4,133
2,356
1,329
768
449
273
164
104
63
40
24
16
9
6
3
2
1
1
162,404
Max T
0
1
193
182
171
160
149
138
127
116
105
94
83
72
61
61
39
30
27
26
25
25
25
25
25
25
193
Min T
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
25
25
25
25
0
Peak T
0
1
18
21
21
21
21
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
Table 2. The number of distinct C∗ -bases, with two divisors D0 and D∞ associated with sections
Σ± connected by N chains of curves of negative self-intersection. Cases N = 0, 1, 2 correspond
to toric bases. P 2 is listed separately. Toric bases with a single 0-curve that can either be a
section (N = 2) or a fiber (N = 1) are listed as N = 1 bases. NSC refers to bases that are not
strictly C∗ -surfaces in that they are described by C∗ -surfaces with -9, -10, or -11 curves in the fiber
chains whose blow-up takes the base outside the set of C∗ -surfaces. The bases described using toric
geometry in [12] all appear in this table, though some are not strictly toric and appear in column 3
and/or in rows with N > 2 due to additional fibers produced when blowing up −9, −10, −11 curves.
Peak T refers to the value of T that occurs for the greatest number of bases at each N . Both NSC
and strictly C∗ -surfaces were used to determine the maximum, minimum, and peak T values.
non-trivial chains, and the number of allowed bases decreases as the number of non-trivial
chains increases above 3.
As mentioned above, the tabulation of bases given in table 2 differs from that of [12]
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JHEP06(2015)061
# Fibers N
P2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
total
– 12 –
JHEP06(2015)061
in the way that toric bases with −9, −10, and −11 curves are treated. In [12], 61,539
bases were identified based on toric structures that in some cases included −9, −10, or −11
curves. These bases include the 34,868 strictly toric bases listed for P2 and N = 0, 1 and
2 in column 2 of table 2, another 8,059 bases that arise from toric bases with −9, −10, or
−11 curves on the fiber chains, included in column 3 of the table, and another 18,612 bases
that are included in rows N > 2 and correspond to C∗ -bases (with or without −9, −10,
or −11 curves on the fiber chains). Bases in the last category arise from toric bases with
−9, −10, −11 curves on the sections D0 , D∞ that give extra nontrivial fibers when the
necessary points are blown up. To summarize, the analysis here includes all the bases
constructed in [12], but discriminates more precisely based on the detailed structure of
these bases.
The parameter T (number of tensor multiplets) serves as the most significant distinguishing characteristic of 6D supergravity theories, and is related to the simplest topological feature h1,1 (B) of the base surface B. Since each nontrivial fiber involves at least one
blowup, and T = 1 for all Fm , T ≥ N + 1. The value of T = h1,1 (B) − 1 for the C∗ -bases
with N > 2 ranges from T = 4 to T = 171. The largest T for any known base is T = 193;
this is an NSC base with only one nontrivial fiber (N = 1). Heuristic arguments were
given in [12] that no other base can have T > 193; as discussed in the next section this
conclusion is supported by the results of this paper. Note that for N from 1 to 13, the
maximum value of T drops by precisely 11 for each increment in N . This can be understood
from the appearance of specific maximal sequences containing −12 curves, as discussed in
section 3.3.
The number of different bases that appear at each T is plotted in figure 5. The number
of bases peaks at T = 25. The distribution of bases also peaks at T = 25 for every specific
N > 5, out to N = 21, 22, 23 and 24, where all possible C∗ -bases have T = 25. One feature
of many of the bases with T = 25 is that they contain precisely two (−12) clusters giving
e8 gauge summands, and no other non-Higgsable clusters. The primary difference between
these bases is that they have different numbers of nontrivial chains and contain different
combinations of intersecting −2 curves. As mentioned above, clusters of −2 curves without
curves of lower self-intersection generally arise at special points in moduli spaces of bases
without such clusters. Indeed, many of the T = 25 bases with gauge algebra e8 ⊕ e8 are
limits of the same complex geometry. We discuss this issue further in section 4.4, in the
context of the full Calabi-Yau threefold geometry of the elliptic fibration over B. In a
similar fashion, many bases have T = 21 and contain one (−12) cluster and a single (−8)
cluster associated with a e7 algebra summand.
The number of C∗ -bases drops off rapidly between about T = 30 and T = 60. Similar
behavior for the toric subset of the bases was noted in [12]. In addition to the primary
peak at T = 25, there are smaller peaks in the distribution of bases starting at T = 50 and
appearing at intervals of eleven, so that they are visible at T = 61, 72, 83, 94, etc. This
feature is also visible when we consider the N -chain cases separately for 3 ≤ N ≤ 14 (see
figure 6). Again, the increment by 11 is related to e8 chain sequences, though the geometry
seems less restricted as T increases.
It may seem surprising that the number of possible bases drops off so rapidly with
Bases
6000
5000
4000
2000
1000
0
0
20
40
60
80
100
120
140
160
180
T
Figure 5. Number of distinct C∗ -bases as a function of the number of tensor multiplets T .
Bases
2500
2000
1500
1000
500
0
0
20
40
60
80
100
120
140
160
180
T
Figure 6. Number of distinct C∗ -bases with N = 3 Fibers (upper blue data) and N = 4 Fibers
(lower purple data) for different numbers T of tensor multiplets.
– 13 –
JHEP06(2015)061
3000
· · ·−12
@
@ −2
−1 @
@
@
@
@
@
@
@ −3
@ −5
@ −3
@ −2
@ −12
−2 @
−1 @
−1 @
−2 @
−1 @
@
@
@
@
@
···
N . Naively, one might imagine a set of K relatively short chains that could be combined
arbitrarily in roughly K N /N ! ways, which could grow rapidly with increasing N for even
modest values of K. The fact that the number of bases decreases rapidly for increasing N
indicates that in fact, not many chains can be combined arbitrarily. These constraints come
in part from the intersection rules that drastically reduce the possibilities of which chains
can simultaneously intersect one of the sections, and in part from the increased number
of blow-ups at points on the sections that are needed to build complicated chains. The
relatively controlled dependence on N of the number of models suggests that going beyond
the class of C∗ -surfaces to completely generic base surfaces may also give a reasonably
controlled number of possible bases.
3.3
Gauge groups and chain structure
We have investigated a number of aspects of the class of C∗ -bases that we have identified.
One of the main conclusions of this investigation is that the structure of the C∗ -bases with
large T is very similar to that of toric bases with large T . For toric bases with large T ,
the intersection structure of divisors with negative self-intersection is largely based on long
chains dominated by sequences of maximal units with gauge algebra e8 ⊕ f4 ⊕ 2(g2 ⊕ su(2))
(See figure 7). The same is true of C∗ -bases. The number of gauge algebra summands of
these types are plotted in figure 8 against T , and grow linearly in a fashion very similar
to that in the toric case. The linear sequences of 12 curves giving the gauge algebra
contributions e8 ⊕ f4 ⊕ 2(g2 ⊕ su(2)) were described in [10]; we refer to these as “e8 units”
for simplicity, as they appear frequently in the intersection structure of bases with large T .
These units are maximal in the sense that they can not be blown up in any way consistent
with the existence of an elliptic fibration.
A detailed analysis of the configurations at large T makes this correspondence clear.
As discussed in [10, 12], the base with T = 193 is a (NSC) base with N = 1, where the
single fiber chain is essentially 16 e8 units ending in −12 curves at D0 , D∞ ; the two −12
curves one e8 unit away from the ends of the fiber are actually −11 curves in the toric base,
which is what makes this base not strictly toric or C∗ (and therefore places it in the third
column of table 2). The N = 2 base with T = 182 is identical except that the long fiber
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JHEP06(2015)061
Figure 7. Bases having large values of h11 (B) = T + 1 have an intersection structure dominated
by multiple repetitions of a characteristic sequence of intersecting divisors associated with nonHiggsable gauge algebra e8 ⊕ f4 ⊕ 2(g2 ⊕ su(2)).
Multiplicity
30
25
20
15
5
0
0
20
40
60
80
100
120
140
160
180
T
Figure 8. Average number of gauge algebra summands as a function of T for summands (ordered
from top at T = 171) g2 ⊕ su(2), e8 , and f4 .
chain contains only 15 e8 units (again with −11 curves one e8 unit away from each end),
and the second chain is a simple (−1, −1) chain arising from a single blow-up on one of the
sections. The unique N = 3 C∗ -base with T = 171 is again similar, with n0 = n∞ = −12,,
two (−1, −1) chains, and the longer chain being a sequence of 14 e8 units, again with two
−11 curves in the C∗ -base. Other bases with large T are dominated in a similar fashion
by one or two very long chains, with the other chains generally being of type (−1, −1) or
other very short chains. For example, there is only one base with N = 3 that has two fiber
chains of length > 50 (see figure 9). For this base there are two long chains of length 53
that combine to form a loop of 9 e8 chains, with a −12 curve and the opposite −5 curve
for D0 and D∞ (and the usual pair of −11 curves near the end on each side), and the third
fiber chain is of the form (−1, −1). Over all N = 3 bases, when the shortest fiber chain
has length > 3 then the next shortest is of length 23 or less.
This analysis of large T C∗ -bases strengthens the conclusion argued heuristically in [12]
that no base can have T > 193. In particular, the C∗ -bases allow two new types of
topological structure to the intersection diagram: loops and branches. Neither of these
types of structure allows for new constructions that qualitatively change the nature of
the bases at large T . In general, bases with large T have a long linear sequence of e8
chains, with additional loops rapidly decreasing the maximum value of T . There are no
new features associated with branches that suggest any mechanism for constructing bases
with large T and more complicated intersection structures. In fact, the known base with
the largest T corresponds to simply a single long e8 chain, with no branching or loops at
all. The only way to modify this structure topologically is to add branchings and loops,
and the results that we have found here show that adding up to two branchings and an
– 15 –
JHEP06(2015)061
10
Longest
Fiber
180
160
140
120
100
60
40
20
0
10
20
30
40
50
Second
Longest
Fiber
Figure 9. Comparison of the lengths of the longest and second longest chains associated with
distinct fibers for bases with N = 3 (blue circles), N = 4 (purple squares), and N = 5 (green stars).
arbitrary number of loops does not give any way of increasing T ; rather, as the topological
complexity increases the upper bound on possible T ’s decreases. Thus, while we still do
not have a proof that T = 193 is the maximum possible for a smooth base B, the absence
of new structure in the C∗ -base models seems to make it unlikely that generic bases with
even more complicated branching and looping structures could have larger values of T ,
even without the C∗ restriction.
4
Calabi-Yau geometry
Over each F-theory base it is possible to construct an elliptically fibered Calabi-Yau geometry. Resolving singularities associated with the nonabelian groups living on the discriminant locus gives a smooth Calabi-Yau threefold. While tuning the moduli over any base
can increase the degree of vanishing of the discriminant locus and hence enhance the nonabelian gauge group, which corresponds to a change in the associated smooth Calabi-Yau
threefold, we focus here primarily on the simplest threefold over each base, corresponding
to the maximally Higgsed 6D supergravity model.
Each Calabi-Yau threefold X has topological invariants given by the Hodge numbers
h1,1 (X), h2,1 (X). A large class of Calabi-Yau threefolds that are realized as hypersurfaces
in toric varieties were identified by Kreuzer and Skarke [9], who produced a comprehensive
database of 474 million Calabi-Yau constructions of this type. These examples include
– 16 –
JHEP06(2015)061
80
Longest
Fiber
180
160
140
120
100
60
40
20
0
Shortest
2
4
6
8
10
12 Fiber
Figure 10. Comparison of the lengths of the longest and shortest chains associated with distinct
fibers for bases with N = 3 (blue circles), N = 4 (purple squares), and N = 5 (green stars).
threefolds with 30,108 distinct pairs of Hodge numbers. The Hodge numbers associated
with allowed toric bases (including some NSC bases) were computed in [13], and it was
shown that the distinctive upper boundary of the “shield region” spanned by the Kreuzer
and Skarke Hodge data is associated with a trajectory of blow-ups of the Hirzebruch surface
F12 . In particular, it was proven that the maximum possible value of h2,1 (X) for any
elliptically-fibered Calabi-Yau threefold (with section) is given by
h2,1 (X) ≤ 491 ,
(4.1)
independent of whether the base is toric or not. In this section we carry out an analysis
of the Hodge numbers for the more general class of C∗ -bases, allowing us to identify some
new Calabi-Yau threefolds with interesting features.
4.1
Calabi-Yau geometry of elliptic fibrations over C∗ -bases
The Hodge numbers of the smooth Calabi-Yau threefold X associated with a generic elliptic
fibration over the base B are related to the intersection structure of the base and to the
gauge group and matter content of the associated 6D supergravity theory.
The Hodge number h1,1 (X) is related to the structure of the base by the Shioda-TateWazir formula [28], which in the language of the 6D supergravity theory states [4, 5]
h1,1 (X) = h1,1 (B) + rank(G) + 1 = T + 2 + rank(Gnonabelian ) + Vabelian ,
– 17 –
(4.2)
JHEP06(2015)061
80
The number of neutral hypermultiplets in the theory on a C∗ -base B can be computed from
the intersection structure on B by analyzing the generic Weierstrass form over that base,
as we describe below in section 4.2. The total number of scalar fields is also related through
the 6D gravitational anomaly cancellation condition to the number of vector multiplets V
in the theory [45, 46]
Hneutral + Hcharged − V = 273 − 29T .
(4.4)
Since the number of charged hypermultiplets can be computed by adding the contributions
from table 1 for each non-Higgsable cluster in B, and the numbers of neutral hypermultiplets, tensors, and nonabelian vectors are also computable from the intersection data
on B, this gives us a way of computing the total number of abelian vector multiplets
Vabelian . Using this in (4.2), we can thus compute both Hodge numbers h1,1 (X), h2,1 (X)
in a systematic way for any C∗ -base.
4.2
Counting neutral hypermultiplets
Computing the number of neutral hypermultiplets over a given base B can be done in a
systematic fashion by following the sequence of blow-ups needed to reach B starting from
a Hirzebruch surface Fm . In the case where B is a toric surface, this analysis is particularly
simple and was described in [12]. For a toric base, the number of neutral hypers is related
to the number of free parameters W in the Weierstrass model for B through
Hneutral = W − waut + N−2
(toric) ,
(4.5)
where waut is the dimension of the automorphism group of B, and N−2 is the number
of curves of self-intersection −2 in B that do not live in clusters carrying a gauge group.
Basically, automorphisms of B correspond to Weierstrass monomials that do not represent
physical degrees of freedom, while −2 curves represent moduli that have been tuned to a
special point, as discussed above. As shown in [12], each curve of self-intersection k ≥ 0
in the toric fan contributes k + 1 to the dimension of the automorphism group, with two
additional universal automorphisms corresponding to the toric structure. For toric surfaces,
– 18 –
JHEP06(2015)061
where G is the full (abelian + nonabelian) gauge group of the 6D theory. While the
nonabelian group is determined as described above from the singularity structure of the
discriminant locus, the rank of the abelian group corresponds to the rank of the MordellWeil group of sections of the fibration. In general, the rank of the Mordell-Weil group is
difficult to compute mathematically (see, e.g., [29]). The Mordell-Weil group is a global
feature of the elliptic fibration, corresponding in the physics context to the number of
U(1) factors in the 6D supergravity theory; the global aspect of this structure is what
makes computation of the group a particularly challenging problem. Recent progress on
understanding abelian factors in general F-theory constructions was made in [30–44].
The Hodge number h2,1 (X) gives the number of complex structure moduli of the
threefold X. This set of moduli represents all but one of the neutral scalar fields in the 6D
gravity theory — the remaining scalar field is associated with the overall Kähler modulus
on the base B.
h2,1 (X) = Hneutral − 1 .
(4.3)
N = 2,
n0 = −2, n∞ = −1
(4.6)
chain 1 : (−1, −3, −1, −2)
(4.7)
chain 2 : (−1, −1)
(4.8)
This base has one (−3) cluster, so the nonabelian gauge algebra is su(3). This base can be
described as F0 with four points blown up, three on the fiber z = 0 and one on the fiber at
– 19 –
JHEP06(2015)061
the number of Weierstrass monomials in f is simply given by the set of points m in the
lattice dual to the toric fan that satisfy hm, vi ≥ −4 for all v in the set of rays generating
the fan, and a similar condition with hm, vi ≥ −6 for monomials in g. As discussed in [12],
there is a simple geometric picture of this in the toric context.
More generally, we can start on Fm with a given number of monomials W in the
Weierstrass degrees of freedom of f, g and explicitly tune the moduli in f, g to blow up the
base at the desired set of points to realize any base B. We can then compute the number of
remaining Weierstrass moduli and apply (4.5) (with one slight modification in the counting
of N−2 as discussed further below). This procedure can be carried out in a clear fashion
when the base B is C∗ . The simplification in the C∗ case comes from the fact that we can
treat each chain arising from a blown-up fiber on Fm in a parallel fashion to the toric case.
In the toric case we are blowing up points on only two fibers, which we can take to be at the
points z = 0, ∞ in a local toric coordinate chart (z, w) ∈ C × C as discussed in section 2.3.
Consider the constraints on Weierstrass monomials coming from blow-ups along the fiber
at z = 0. Each such constraint corresponds to imposing a vanishing condition on a set
P
of monomial coefficients in an expansion f = n,m cn,m z n wm , and similarly for g. The
set of constraints imposed depends upon the sequence of blow-ups in the fiber, and has a
convenient geometric description in the language of the toric fan and dual space. For a
C∗ -base with N chains located at z = z1 , . . . , zN , we simply impose the appropriate set of
P
(i)
constraints on the monomial coefficients in an expansion f = n,m cn,m (z − zi )n wm for the
ith chain. This gives a set of linear conditions on the coefficients in f, g. The dimension of
the space of independent solutions to these linear constraints can then be used to compute
the number of independent Weierstrass monomials W , in an analogous fashion to (4.5). As
in the toric case, when one of the sections D0 , D∞ has self-intersection k ≥ 0 it contributes
k + 1 to the dimension of the automorphism group. For a C∗ -base with N = 3 or more
blown up fibers, the first two fibers can be fixed at z = 0, ∞, using up two automorphisms
originally associated with the fibers of self-intersection 0 in the toric base. The blowing
up of the third fiber corresponds to fixing another point in C∗ , which can for example be
chosen to be z = 1, fixing one of the remaining automorphisms and leaving only one of
the universal two present in all toric bases. For each of the N − 3 fibers beyond the third,
blowing up an additional point involves choosing a value zi which is itself a Weierstrass
parameter though it does not correspond to a monomial. Thus, the formula (4.5) must be
augmented by N − 3 when more than 3 fibers are blown up. The complete formula for the
number of neutral hypermultiplets is given at the end of this section, in (4.12).
It may be helpful to illustrate this method with an example. Consider first the toric
base B with the following parameters
z = ∞. A graphic description of the toric fan and monomials in the dual lattice is given
in figure 11.
The monomials in this toric model can be computed using the methods of [12]. The
monomials in f in the Weierstrass model on F0 are of the form cn,m z n wm , 0 ≤ n, m ≤ 8,
P
n m
and similarly for g = 12
n,m=0 dn,m z w . Blowing up the points on F0 imposes conditions
on the coefficients c, d. For example, blowing up the point z = w = 0 (corresponding
to adding the ray in red in the diagram on the right-hand side of figure 11) imposes the
conditions that cn,m = 0 for all n + m < 4 and dn,m = 0 for n + m < 6. This removes 31
Weierstrass moduli (and reduces the automorphism group by 2 by turning two 0-curves into
−1-curves), giving a change in the number of neutral hypermultiplets ∆Hneutral = −29,
matching (4.4) with ∆T = 1. This corresponds to removing all the monomials on the
lower-left corner of the left-hand diagram in figure 11. The analogous constraints imposed
by blowing up the other points on the fiber z = 0 as well as the point z = ∞, w = 0
are depicted in an analogous fashion in the figure. Together, these constraints reduce the
number of Weierstrass monomials to W = 136. This matches with (4.5) and (4.4), with
N−2 = waut = 2 and V = 8 vector multiplets from the su(3) factor in the gauge algebra.
Thus, we have determined the Hodge numbers of the resolved generic elliptic fibration
over B
h1,1 (X) = 9,
h2,1 (X) = 135 .
(4.9)
Now, consider blowing up a point on a third fiber at z = 1 to form the C∗ -base B 0
shown in figure 12. Blowing up at the point (z, w) = (1, 0) imposes the condition that f
and g must vanish to degrees 4 and 6 in (z − 1) and w. Consider for example f . From the
geometry depicted in figure 11 it is clear that the only undetermined coefficients of f at
– 20 –
JHEP06(2015)061
Figure 11. An example of counting monomials in a toric base B. The base F0 has 250 Weierstrass
coefficients: 9 × 9 = 81 in f (solid dots) and 13 × 13 in g (round circles) in the dual lattice. These
are all of the dots and circles in the diagram on the left. Each blow-up corresponds to adding a
vector to the fan (above right), which removes some monomials from f and g. For example, blowing
up the point corresponding to the red ray in the diagram on the right removes all monomials below
and to the left of the red line in the left-hand diagram, blowing up on the blue ray removes all
points above and to the left of the blue line, etc.. The base B has 136 monomials, 44 in f and 92 in
g, corresponding to the dots and circles in the left-hand diagram that lie above the red and green
lines and to the right of the purple line.
D0
-2
D1,1 -1
B
B
D1,3 -1
B
D1,4 B -2
B
D1,1 -1
D2,1 -1
B
D1,2 B -3
0
B
-
D2,2 BB -1
B
B
B
B
B
D1,4 B -2
B
D∞
D3,1 -1
D1,3 -1
B
-1
D2,1 -1
D1,2 B -3
D0
-3
B
B
B
D2,2 BB -1
D3,2 BB -1
-1
B
B
B
B
D∞
leading orders in w are
f = c4,0 z 4 + c3,1 z 3 w + c4,1 z 4 w + c5,1 z 5 w +
6
X
cn,2 z n w2 + · · · .
(4.10)
n=2
The condition that this vanishes to degree 4 in (z − 1), w forces all coefficients cn,0 and cn,1
to vanish, imposes two constraints at m = 2, and one constraint at m = 3, totaling 7 new
constraints. Similarly, coefficients of g experience 14 further constraints, for a total of 21
constraints. This reduces the number of Weierstrass monomials to W = 115. The equations
(4.5) and (4.4) still apply, where now waut = 1 since one of the toric automorphisms is
broken by the reduction to C∗ structure, and N−2 = 1 because the blow-up at z = 1, w = 0
changes n0 from −2 to −3. With one new −3 curve the number of vector multiplets becomes
16, and the resolved generic elliptically fibered Calabi-Yau X 0 over B 0 has Hodge numbers
h1,1 (X 0 ) = 12,
h2,1 (X 0 ) = 114 .
(4.11)
In this way we can determine the Hodge numbers of the Calabi-Yau threefolds associated with generic elliptic fibrations over all C∗ -bases. When the additional fibers added are
more complicated, the conditions on f, g at z = zi can be determined by simply translating
the conditions at z = 0 from the toric picture.
The one remaining subtlety in this general picture is that for general C∗ -bases some
combinations of −2 curves must be treated specially.4 In particular, for certain configurations of intersecting −2 curves associated with Kodaira-type surface singularities there
is a linear combination that describes a degenerate genus one curve [20]. In these cases,
the extra deformation directions associated with the −2 curves are not independent and
the contribution from N−2 to (4.5) is reduced by 1. The −2 curve configurations of these
types that appear in C∗ -bases are shown in figure 13. These −2 curve configurations can be
identified as those where an integral linear combination of the −2 curves is a divisor with
vanishing self-intersection. For this to occur, the weighting of any −2 curve Ci must be 1/2
the total of the weightings of the −2 curves that intersect Ci . Some simple combinatorics
4
Thanks to David Morrison for discussions on this point
– 21 –
JHEP06(2015)061
Figure 12. An example of a C∗ -surface B 0 given by blowing up the toric surface B from figure 11
at a point.
C
C
C
C
1 C
1 1
(a)
2
1 1 C
1 C
C
C
2 I0∗ (D̂4 )
1 C
C
C
2
C
C
3
(b) IV ∗ (Ê6 )
2 CC
1 1 2
C
2C
C
C
C
2 C
2 CC
2 CC
C
C
3 C
3
4
(c) III ∗ (Ê7 )
C
3 4 CC
C
4C
5
6
(d) II ∗ (Ê8 )
C
3 CC
Figure 13. Configurations of −2 curves associated with Kodaira-type surface singularities associated with degenerate elliptic fibers. For these configurations, the number of fixed moduli associated
with −2 curves is reduced by one. The numbers given are the weightings needed to give an elliptic
curve with vanishing self-intersection. Labels correspond to Kodaira singularity type and associated
Dynkin diagram.
shows that the configurations in figure 13 are the only possible geometries satisfying this
condition that have a single −2 curve that intersects more than two others.5 Thus, for
C∗ -bases that are not toric (N ≥ 3), the formula (4.5) is replaced by
Hneutral = W − waut + (N − 3) + N−2 − G1
(C∗ ) ,
(4.12)
where waut = 1 + max(0, 1 + n0 , 1 + n∞ ), N−2 is the number of -2 curves, and G1 is the
number of −2 configurations of the types shown in figure 13.
4.3
Distribution of Hodge numbers
We have computed the Hodge numbers for the generic threefolds over all 162,404 C∗ -bases,
including those not strictly C∗ -bases coming from blown up C∗ -bases with −9, −10, and
−11 curves on the fiber chains. We find a total of 7,868 distinct pairs of Hodge numbers
h1,1 , h2,1 , including 344 Hodge number combinations not found in [13] from the set of
generalized toric bases. These Hodge numbers are plotted and compared to the KreuzerSkarke database [9] and the Hodge numbers of threefolds associated with toric bases in
One other class of configurations satisfies this condition, the type Ib∗ singularity, with two −2 curves
each intersecting three −2 curves and connected by a single chain of −2 curves, but this cannot appear in
a C∗ -base since the chain associated with each fiber must contain at least one −1 curve, and in this case
the chain connecting the two -2 curves with triple branching contains itself only −2 curves.
5
– 22 –
JHEP06(2015)061
1 500
h2,1
400
300
100
0
0
100
200
300
400
500
h1,1
Figure 14. Plot of Hodge numbers comparing C∗ (shown in dark blue, largest dots), KreuzerSkarke (slate blue, medium dots), and toric (green, smallest dots). Hodge numbers for C∗ -bases that
are not also Hodge numbers for toric (or NS-toric) bases are generally those with small h1,1 , h2,1
with a small number of exceptions having larger Hodge numbers, including 6 examples that are not
found in the Kreuzer-Skarke database.
figure 14. The number of models as a function of the sum h1,1 + h2,1 is plotted in figure 15.
Most of the new Hodge numbers that appear for C∗ -bases and not for toric bases are in
the region of small Hodge numbers far from the boundary.
Note that just as for toric bases, as discussed in [12], the set of Calabi-Yau manifolds
that can be constructed over any given C∗ -base B can be very large. By tuning parameters
in the Weierstrass model over any given base, so that f and g vanish on certain divisors
to degrees less than 4, 6, theories with many different nonabelian gauge group factors
can be constructed. Each such construction gives a different Calabi-Yau threefold after
the singularities in the elliptic fibration are resolved. In this way, the number of Hodge
numbers associated with Calabi-Yau threefolds fibered over any given C∗ -base can be quite
large. Explicit examples of such tunings that give Hodge numbers near the boundary of
the “shield” for threefolds over toric bases are described in [13, 47]. Similar constructions
over C∗ -bases would give a vast range of different Calabi-Yau threefold constructions.
4.4
Redundancies from −2 clusters
One striking feature of the distribution of Hodge numbers is that there are certain Hodge
number combinations that are realized by the threefolds associated with a large number of
distinct C∗ -bases. As the most extreme example, there are 1,861 different C∗ -bases with
Hodge numbers 43, 43. Many of these are in fact just different realizations of the same
Calabi-Yau threefold.
– 23 –
JHEP06(2015)061
200
Bases
8000
7000
6000
5000
4000
2000
1000
0
0
100
200
300
400
500
h1,1 + h2,1
Figure 15. Distribution of threefolds coming from generic elliptic fibrations over C∗ -bases as a
function of the sum of Hodge numbers.
One principal source of these kinds of redundancies arises from the appearance of
clusters of −2 curves in the base that do not carry a gauge group. As discussed previously,
such −2 curves indicate that a modulus of the geometry has been tuned. In general, we
expect that any base containing a cluster of −2 curves is just be a special limit of another
base without such clusters. Thus, we can consider the subset of C∗ -bases that do not
contain any clusters of only −2 curves. This reduces the number of bases to 68,798. A
graph of the distribution of the numbers of bases without −2 clusters as a function of T is
given in figure 16.
The removal of −2 clusters removes a great deal of redundancy in the list of threefolds.
In particular, in the reduced set of bases the extreme jump in the distribution at T = 25
goes away. This is associated with the removal of a large number of threefolds with Hodge
numbers (43, 43). Looking at the detailed data shows that many of these (43, 43) models
have a closely related structure. There are 1,575 C∗ -bases that have the following features:
• Two (−12) clusters and a gauge algebra e8 ⊕ e8
• T = 25
• h1,1 = 43, h2,1 = 43 .
In general, these bases are characterized by e8 factors on the sections (n0,∞ = −12),
and a set of chains containing various combinations of −2 curves. Indeed, it is clear
that there are many ways to construct such C∗ -bases, by starting with a given Fm and
– 24 –
JHEP06(2015)061
3000
Bases
3000
2500
2000
1500
500
0
0
50
100
150
200
T
Figure 16. The number of C∗ -bases associated with different values of T = h1,1 (B) − 1, when
only bases without clusters of −2 curves not carried a gauge group are considered.
blowing up only points at w = 0, ∞. Such C∗ -bases will always have chains of the form
(−1, −2, −2, . . . , −2, −1). If precisely 24 points are blown up, giving the necessary factors
on the sections, then any combination of chain lengths satisfying
N
X
(ki − 1) = 24 = T − 1
(4.13)
i=1
will have the desired properties. The number of such bases is just the number of ways of
partitioning the 24 blow-ups into a sum of integers, p(24) = 1, 575, precisely the number
of bases found with the above features. Of these partitionings, 13 have toric descriptions
(partitions into one or two integers). All 1575 C∗ -bases in this set can be understood
as limit points of the same geometry, and are associated with the same smooth CalabiYau threefold. This Calabi-Yau threefold, which has been encountered previously in the
literature (see e.g. [25]), seems to have a particularly high degree of symmetry and may be
interesting for other reasons.
Although many of the bases with clusters of −2 curves have the same Hodge numbers as
bases without such clusters, this is not universally true. There are also 3,788 C∗ bases that
have −2 clusters that have no corresponding C∗ -base without such clusters. An example
is given by base B with
N = 4,
n0 = −2, n∞ = −4
chain 1 : (−2, −1, −4, −1, −3, −1)
chains 2 − 4 : (−1, −1)
– 25 –
JHEP06(2015)061
1000
4.5
Calabi-Yau threefolds with new Hodge numbers
Of the roughly 160,000 C∗ -bases, we have found precisely 6 that give rise to generic elliptic
fibrations with Hodge numbers that are not found in the Kreuzer-Skarke database. The
simplest such base has Hodge numbers h1,1 = 56, h2,1 = 2, and the structure
N = 3, n0 = −5, n∞ = −6,
(4.14)
chain 1 : (−1, −3, −1, −3, −1)
chain 2 : (−1, −3, −2, −1, −5, −1, −3, −1)
chain 3 : (−1, −3, −2, −2, −1, −6, −1, −3, −1)
The other 5 bases that give new Calabi-Yau threefolds are listed in appendix B.
4.6
Calabi-Yau threefolds with nontrivial Mordell-Weil rank
There are 13 C∗ -bases in which the rank of the Mordell-Weil group is nonzero. This is determined, as described above, by using the monomial count and the anomaly equation to
independently determine Hneutral and Hneutral − Vabelian for each of the C∗ -bases. The bases
where these two quantities differ are those for which the Mordell-Weil rank is nonzero. The
elliptically fibered Calabi-Yau threefolds over these bases have multiple linearly independent sections in a group of rank r. This gives rise to r abelian U(1) gauge fields in the
corresponding 6D supergravity theory.
For these 13 bases, therefore, the Mordell-Weil rank is forced to be nontrivial even for
a completely generic elliptic fibration. This is different from the situation for toric bases,
where a complete analysis of all toric bases using the anomaly condition (4.4) confirmed
that there are no toric bases over which the generic elliptic fibration has a nontrivial
Mordell-Weil rank.
– 26 –
JHEP06(2015)061
These bases may be limits of other base surfaces that are fine as complex surfaces but do
not have a description as a C∗ -surface.
While as discussed above, in general bases with −2 clusters correspond to limits of
bases without such clusters, and amounts to a redundancy for generic elliptic fibrations,
keeping track of this information is often useful. In particular, Weierstrass models over a
base with −2 clusters can be tuned to realize smooth resolved Calabi-Yau threefolds that
cannot be realized by tuned Weierstrass models over the more generic bases without the −2
clusters. Thus, bases with −2 clusters must be considered separately in any systematic or
complete analysis of a set of Calabi-Yau threefolds; examples of this arise in [47]. In terms
of the physical F-theory models that can be constructed from these bases, the different −2
configurations and the distinct configurations of tuned gauge groups that can be realized
over them correspond to distinct classes of 6D supergravity theories with distinct gauge
group, matter, and dyonic string lattice structure. Thus, in a full consideration of 6D
supergravity theories, it would be necessary to include all of the distinct base choices with
Hodge numbers (43, 43) as initial points for tuned models with different sets of Hodge
numbers.
An example of a base with nonzero Mordell-Weil rank is given by the following base
with Hodge numbers h1,1 = 25, h2,1 = 13:
N = 3, n0 = −1, n∞ = −2,
(4.15)
chain 1 : (−2, −1, −2)
chain 2 : (−4, −1, −2, −2, −2)
chain 3 : (−4, −1, −2, −2, −2)
V = Hneutral + 29T − 273 = 60 = 56 + r .
(4.16)
Thus, this base has a generic elliptic fibration with Mordell-Weil rank r = 4, and h1,1 = 25.
The 13 examples of bases with Mordell-Weil groups having nonzero rank are listed in
appendix C
The appearance of such bases, while perhaps surprising, is not completely unprecedented. It is known that the Shoen manifold, a class of elliptically fibered Calabi-Yau
threefold constructed from a fiber product of rational elliptic surfaces, generically has
Mordell-Weil rank 9 [48] (see e.g. [49] for a physics application of this). In fact, all the
C∗ -bases we have identified that give Calabi-Yau threefolds with enhanced Mordell-Weil
rank can be related to special limits and blow-ups of the Shoen manifold;6 this connection
will be described in further detail elsewhere [50].
For the bases with nonzero Mordell-Weil rank, an explicit description of the Weierstrass
model can be used to make the extra sections manifest. As an example, consider the base
N = 4, n0 = −6, n∞ = −6,
(4.17)
chain 1 : (−1, −3, −1, −3, −1)
chain 2 : (−1, −3, −1, −3, −1)
chain 3 : (−1, −3, −1, −3, −1)
chain 4 : (−1, −3, −1, −3, −1)
This base has T = 17, h1,1 = 51, h2,1 = 3, and Mordell-Weil rank r = 4. An explicit
computation of the monomials in the Weierstrass model gives (placing the extra two fibers
at z = 1, 2)
f = Aw4 z 2 (z − 1)2 (z − 2)2
4 6
6
(4.18)
6
6 3
3
3
8
g = Bw z (z − 1) (z − 2) + Cw z (z − 1) (z − 2) + Dw ,
6
Thanks to David Morrison for discussions on this point.
– 27 –
JHEP06(2015)061
The Hodge number h1,1 can be computed from this base using (4.2), and T = 11 from
(2.2), where G = so(8) ⊕ so(8) from the two −4 curves, so h1,1 (X) = T + 2 + 8 + r = 21 + r,
where r is the rank of the Mordell-Weil group. Using the method of section 4.2, the
number of Weierstrass monomials can be computed to be W = 7. The dimension of the
automorphism group is the generic waut = 1, and N−2 = 9 is the number of −2 curves,
with G1 = 1 as described at the end of section 4.2 since the −2 curves connected to D∞
have the IV ∗ (Ê6 ) form from figure 13. It follows then from (4.12) that Hneutral = 14, so
h2,1 = 13. Comparing with (4.4), we have
where A, B, C, D are free complex constants (W = 4). A nontrivial section can be associated with a factorization of the Weierstrass equation y 2 = x3 + f x + g into the form [33]
(y − α)(y + α) = (x − λ)(x2 + λx − µ)
(4.19)
f = −µ − λ2
(4.20)
g = λµ + α2 .
(4.21)
where
λ = aw2 z(z − 1)(z − 2)
4 2
2
µ = bw z (z − 1) (z − 2)
(4.22)
2
(4.23)
α = cw2 z 3 (z − 1)3 (z − 2)3 + dw4
(4.24)
A = −a2 − b
(4.25)
2
(4.26)
C = ab + 2cd
(4.27)
2
(4.28)
where a, b, c, d satisfy
B =c
D=d .
For given generic values of A-D there are 12 solutions for a-d. This can be seen by noting
that the equations for B, D each have two solutions for c, d, and the other two equations
combine to form a cubic for a, which has 3 independent solutions. These twelve solutions
represent 4 independent generators of the Mordell-Weil group. Solutions with α → −α
correspond to sections s, −s that add to 0. The three solutions for the cubic also contain
one linear dependence in the space of sections, so that the total number of independent
sections is 4, matching the computed Mordell-Weil rank. A similar computation can be
carried out for the other bases with enhanced Mordell-Weil rank, though the details are
more complicated and computing the number of independent sections can be more difficult
in other cases. A more detailed analysis of these models with enhanced Mordell-Weil rank
is left to future work.
5
Conclusions
In this paper we have initiated a systematic study of a class of geometries for the bases of
elliptically fibered Calabi-Yau threefolds that goes beyond the framework of toric geometry
widely used in previous work. We have systematically constructed all smooth surfaces that
admit a single C∗ action and can arise as bases of a Calabi-Yau threefold, and we have analyzed the properties of these geometries. The 162,404 bases we have explicitly constructed
include all C∗ -bases and also a more general class built from C∗ -bases containing −9, −10,
and −11 curves on which points must be blown up to form −12 curves in bases that do
not have a C∗ action.
– 28 –
JHEP06(2015)061
The Weierstrass coefficients from (4.18) can take this form if
– 29 –
JHEP06(2015)061
The bases we have considered can be used for compactification of F-theory to six
dimensions. We have found that the physical properties of the resulting six-dimensional
supergravity theories are similar in nature and in distribution to compactifications on toric
bases that were studied earlier. In particular, the C∗ -bases with relatively large numbers
T of tensor multiplets give theories with gauge algebras that are dominated by summands
of the form e8 ⊕ f4 ⊕ 2(g2 ⊕ su(2)), from the same types of chains that give this structure
in the toric case. The largest value of T for theories with C∗ -bases that are not actually
toric is 171, lower than the largest known value of T = 193 that occurs for a toric base
(in both cases, these are examples of bases containing −11 curves that must be blown
up to give bases outside the C∗ /toric framework). The overall “shield” structure of the
set of Hodge numbers computed by Kreuzer and Skarke based on toric constructions, the
boundary of this region explained in [12], and geometric patterns identified in [12, 51]
are essentially unchanged with the addition of the large number of more general non-toric
C∗ -base configurations. The absence of branching or loop structures in the C∗ -bases that
make possible higher values of T supports the conclusion that T ≤ 193 is an absolute bound
across all bases, as was argued heuristically in [12].
One interesting result of this analysis is that the number of additional bases added by
extending the toric construction to include the more general class of bases admitting a C∗
action does not produce a wild increase in the number of possible bases. As more points are
blown up, the set of geometries is generally controlled by linear structures along the fiber,
and the possibility of branching does not lead to a combinatorial explosion in intersecting
divisor structures. This suggests that a systematic classification of all smooth bases for
elliptically fibered Calabi-Yau threefolds, even without a single C∗ action, may be computationally tractable. Such a classification would be quite challenging, since the intersection
structure can become quite complicated — particularly for T ≥ 9 where there can be an
infinite number of distinct −1 curves on the bases, such as occurs for dP9 . Nonetheless, by
using the method of non-Higgsable clusters to characterize possible geometries it may be
possible to get a handle on this problem. We leave this as a challenge for future work.
We have analyzed the Hodge structure of the smooth Calabi-Yau threefolds associated
with generic elliptic fibrations over all the C∗ -bases. While the Hodge numbers are all
within the boundaries defined by toric bases and the Kreuzer-Skarke “shield” shape, we
have identified a number of Calabi-Yau threefolds with novel properties. There are 6 new
threefolds that have Hodge numbers that we believe have not been previously identified, as
well as 13 threefolds in which the Mordell-Weil rank of the elliptic fibration is nontrivial,
corresponding to non-Higgsable U(1) factors in the corresponding 6D supergravity theory.
The basic approach to constructing a more general class of base manifolds for elliptic
fibrations and F-theory using spaces with reduced numbers of C∗ actions should be possible
for base threefolds and fourfolds as well, giving rise to geometries for compactification of
F-theory and M-theory to dimensions from 5 to 2. The analysis of these constructions
becomes significantly more complicated in lower dimensions, however. In particular, for
F-theory compactifications to four dimensions on elliptically-fibered Calabi-Yau fourfolds
with threefold bases, the analysis of bases with a C∗ × C∗ or C∗ action would be much
more difficult than for the class of C∗ -surfaces considered here. One difficulty in analyzing
such constructions is the absence of a clear analogue to the anomaly condition (4.4) for 6D
theories. The absence of such a condition makes it harder to compute the Mordell-Weil
group in a general 4D case, and hence more difficult to precisely identify the Hodge numbers
of the elliptically-fibered fourfold. While some progress was made in [52] in identifying 4D
parallels to the 6D anomaly condition and related structures, more work is needed to have
a systematic approach to analyzing F-theory base spaces with reduced toric symmetry in
the 4D case.
We would like to thank Lara Anderson, Philip Candelas, Mboyo Esole, Ilarion Melnikov,
David Morrison, Daniel Park, Tony Pantev, and Yinan Wang for helpful discussions. This
research was supported by the DOE under contract #DE-FC02-94ER40818, and was also
supported in part by the National Science Foundation under Grant No. PHY-1066293. WT
would like to thank the Aspen Center for Physics for hospitality during part of this work.
A
Rules for connecting clusters in C∗ -bases
All configurations of non-Higgsable clusters (NHC’s) that can be connected by a single −1
curve in a C∗ -base are listed in table 3. Some examples of how some of these configurations
can arise in C∗ -bases are given in figure 17.
There are a few additional constraints not analyzed in [10], that arise from conditions
on branching over a −n curve with n > 1; these additional constraints arise for similar
reasons to those underlying the constraints on branching over a −1 curve. The additional
constraints that must be imposed on such branchings are as follows:
• −3 curves:
A −3 curve that is connected to a −2 curve can be connected to at most one chain
that begins −1, −5, . . ..
• −5 curves:
A −5 curve can connect to at most two chains that begin −1, −3, −2, . . ..
• −7 curves:
A −7 curve can connect to at most five chains that begin −1, −2, −3, . . ..
• −8 curves:
A −8 curve can connect to at most four chains that begin −1, −2, −3, . . ..
• −12 curves:
A −12 curve can connect to at most eight chains that begin −1, −2, −2, −3, . . ..
– 30 –
JHEP06(2015)061
Acknowledgments
NHC
˙
(−12)
(−8̇)
(−7̇)
(−6̇)
(−5̇)
(−4̇)
(−3̇)
(−3̇, −2)
(−3, −2̇)
(−3̇, −2, −2)
(−3, −2̇, −2)
(−3, −2, −2̇)
(−2̇, −3, −2)
(−2, −3̇, −2)
pairs {triplets [quadruplets]}
12
9, 12, 13
9, 12, 13
7, 9, 11, 12, 13
7, 8, 9, 10, 11, 12, 13
6, 7, 8, 9 {12}, 10, 11, 12 {12, 13}, 13, 14
7 {7, 9, 12}, 8, 9 {9, 12, 13}, 10, 11, 12 {12, 13}, 13 {13}, 14
8, 9 {9, 12}, 10, 11, 12 {12, 13}, 13, 14
9 {9, 10, 12 [12], 13}, 10 {12}, 11, 12 {12 [12, 13], 13}, 13 {13}, 14
10, 11, 12 {12, 13}, 13, 14
12 {12, 13}, 13 {13}
12 {12 [12, 13], 13 [13]}, 13 {13}, 14
13 {13}, 14
Table 3. Table of all ways in which connections between non-Higgsable clusters by (-1)-curves
are allowed to appear in a C∗ -surface that can be used for an F-theory base. The dots specify the
curves within each NHC that intersect with the (-1)-curve. The last column provides a complete list
of the allowed pairs, triplets and quadruplets of NHCs that can be intersected by a single (-1)-curve
in a valid F-theory base. No more than 4 NHCs can be connected by a single (-1) curve. Note also
that any of the single NHCs or set of several NHCs that can be intersected by a single (-1)-curve
can also be connected by the same (-1)-curve to sets of -2 curves, which do not carry a nonabelian
gauge group. This table contains the subset of the data from an analogous table in [10] that is
relevant for C∗ -bases. Note that combinations are listed only in increasing numerical order, so that
a given combination only appears on the row associated with the lowest numbered NHC in the
connected set.
As an example of how these constraints arise, consider the Zariski decomposition of
−4K, −6K for a sequence of rational curves A, B, C, D of self-intersections −5, −1, −3, −2
with mutual intersections A · B = B · C = C · D. From the relations −K · A = −3, −K · C =
−1, . . . we have
−4K = 3A + B + 2C + D + X
(A.1)
−6K = 4A + B + 3C + 2D + Y
(A.2)
where X, Y are residual effective divisors that have nonnegative intersection with each of
the curves A-D. This Zariski decomposition is compatible with the (3, 4) vanishing on the
−5 curve, etc., and implies that (f, g) both vanish to order one on the curve B. Now,
consider a configuration where the curve A of self-intersection −5 is attached to three
sequences (B, C, D), (B 0 , C 0 , D0 ), (B 00 , C 00 , D00 ) of curves of self-intersections −1, −3, −2.
Then both −4K and −6K would have to contain all of B, B 0 , B 00 as components. Since,
however, (4A + B + B 0 + B 00 ) · A = −17, while −6K · A = −18, the Zariski decomposition of
−6K must contain at least 5A, so that g must vanish to order 5 on A. This would mean,
however, that (f, g) would vanish to orders (4, 6) at the point A ∩ B, which we do not
– 31 –
JHEP06(2015)061
Index (n)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
C
C
C
C
C
-6 C
-1 C
C
C
C
C
-1 -2
C
C
C
-1 C
-1 C
C
-2 C
C
C
C
-2 C
-4 C
C
-3 C
-1 (a)
-3
-1 C
C
C
C
C
-2 (b)
allow. This demonstrates the condition stated above for branching on a −5 curve. Similar
arguments can be used to show the other conditions.
The conditions on −3 and −5 curve branching rule out some additional candidates for
∗
C -bases that are otherwise apparently acceptable.7
B
Bases giving Calabi-Yau threefolds with new Hodge numbers
In addition to the example (4.14) given in section 4.5, there are 5 other C∗ -bases over which
the generic elliptic fibration has Hodge numbers not found in the Kreuzer-Skarke database:
Hodge numbers h1,1 = 61, h2,1 = 1:
N = 3, n0 = −5, n∞ = −6,
(B.1)
chain 1 : (−1, −3, −1, −3, −1)
chain 2 : (−1, −3, −2, −2, −1, −6, −1, −3, −1)
chain 3 : (−1, −3, −2, −2, −1, −6, −1, −3, −1)
Hodge numbers h1,1 = 62, h2,1 = 2:
N = 3, n0 = −4, n∞ = −8,
(B.2)
chain 1 : (−1, −4, −1, −2, −3, −2, −1)
chain 2 : (−1, −4, −1, −2, −3, −2, −1)
chain 3 : (−1, −4, −1, −2, −3, −2, −1)
Hodge numbers h1,1 = 110, h2,1 = 2:
N = 3,
n0 = −6, n∞ = −12,
(B.3)
chain 1 : (−1, −3, −1, −3, −2, −2, −1)
chain 2 : (−1, −3, −1, −3, −2, −2, −1)
chain 3 : (−1, −3, −1, −6, −1, −3, −1, −6, −1, −3, −1, −5, −1, −3, −2, −2,
−1, −12, −1, −2, −2, −3, −1, −5, −1, −3, −2, −2, −1)
7
These additional rules were missed in an earlier version of this paper, which led to the appearance
of four additional candidate C∗ -bases. Remarkably, these four all were in the class of bases with generic
nonzero rank for the Mordell-Weil group. Thanks to D. Morrison, D. Park, and Y. Wang for discussions
on related points.
– 32 –
JHEP06(2015)061
Figure 17. Examples of how clusters connected by −1 curves can arise at the branching points
of the configuration of divisors in C∗ -bases.
Hodge numbers h1,1 = 241, h2,1 = 31:
N = 3,
n0 = −1, n∞ = −5,
(B.4)
chain 1 : (−1, −1)
chain 2 : (−2, −1, −3, −1)
chain 3 : −12// − 11// − 12// − 12// − 12// − 12// − 12// − 11, −1, −2, −2, −3, −1)
where, as in [13], the notation X//Y denotes the sequence X, −1, −2, −2, −3, −1, −5, −1,
− 3, −2, −2, −1, Y .
N = 3,
n0 = −1, n∞ = −6,
(B.5)
chain 1 : (−1, −1)
chain 2 : (−2, −1, −3, −1)
chain 3 : −1, −12// − 11// − 12// − 12// − 12// − 12// − 12// − 12// − 1, −2, −2, −3,
−1, −5, −1, −3, −1)
C
Bases giving U(1) factors
This appendix contains a list of the 13 bases over which a generic elliptic fibration has
Mordell-Weil group of nonzero rank r. Note that two of these (numbers 4, 5) also appear
in the list of Calabi-Yau’s with new Hodge numbers.
r = 1 : N = 3, n0 = −2, n∞ = −3; T = 15; h1,1 = 34, h2,1 = 10
(C.1)
chain 1 : (−2, −1, −2)
chain 2 : (−2, −2, −1, −4, −1)
chain 3 : (−2, −2, −2, −2, −2 − 1, −8, −1, −2)
r = 2 : N = 3, n0 = −1, n∞ = −2; T = 12; h1,1 = 24, h2,1 = 12
(C.2)
chain 1 : (−2, −1, −2)
chain 2 : (−3, −1, −2, −2)
chain 3 : (−6, −1, −2, −2, −2, −2, −2)
r = 2 : N = 3, n0 = −2, n∞ = −6; T = 18; h1,1 = 46, h2,1 = 10
chain 1 : (−2, −1, −3, −1)
chain 2 : (−2, −2, −2, −1, −6, −1, −3, −1)
chain 3 : (−2, −2, −2, −1, −6, −1, −3, −1)
– 33 –
(C.3)
JHEP06(2015)061
Hodge numbers h1,1 = 252, h2,1 = 30:
r = 2 : N = 3, n0 = −5, n∞ = −6; T = 21; h1,1 = 61, h2,1 = 1
(C.4)
chain 1 : (−1, −3, −1, −3, −1)
chain 2 : (−1, −3, −2, −2, −1, −6, −1, −3, −1)
chain 3 : (−1, −3, −2, −2, −1, −6, −1, −3, −1)
r = 3 : N = 3, n0 = −4, n∞ = −8; T = 19; h1,1 = 62, h2,1 = 2
(C.5)
chain 1 : (−1, −4, −1, −2, −3, −2, −1)
chain 3 : (−1, −4, −1, −2, −3, −2, −1)
r = 4 : N = 3, n0 = −1, n∞ = −2; T = 11; h1,1 = 25, h2,1 = 13
(C.6)
chain 1 : (−2, −1, −2)
chain 2 : (−4, −1, −2, −2, −2)
chain 3 : (−4, −1, −2, −2, −2)
r = 4 : N = 3, n0 = −1, n∞ = −5 T = 14; h1,1 = 40, h2,1 = 4
(C.7)
chain 1 : (−2, −1, −3, −1)
chain 2 : (−4, −1, −2, −2, −3, −1)
chain 3 : (−4, −1, −2, −2, −3, −1)
r = 4 : N = 4, n0 = −6, n∞ = −6; T = 17; h1,1 = 51, h2,1 = 3
(C.8)
chain 1 : (−1, −3, −1, −3, −1)
chain 2 : (−1, −3, −1, −3, −1)
chain 3 : (−1, −3, −1, −3, −1)
chain 4 : (−1, −3, −1, −3, −1)
r = 4 : N = 3, n0 = −2, n∞ = −4; T = 13; h1,1 = 35, h2,1 = 11
(C.9)
chain 1 : (−2, −2, −1, −4, −1)
chain 2 : (−2, −2, −1, −4, −1)
chain 3 : (−2, −2, −1, −4, −1)
r = 5 : N = 3, n0 = −1, n∞ = −8; T = 16; h1,1 = 51, h2,1 = 3
chain 1 : (−3, −1, −2, −3, −2, −1)
chain 2 : (−3, −1, −2, −3, −2, −1)
chain 3 : (−3, −1, −2, −3, −2, −1)
– 34 –
(C.10)
JHEP06(2015)061
chain 2 : (−1, −4, −1, −2, −3, −2, −1)
r = 6 : N = 3, n0 = −1, n∞ = −2; T = 10; h1,1 = 24, h2,1 = 12
(C.11)
chain 1 : (−3, −1, −2, −2)
chain 2 : (−3, −1, −2, −2)
chain 3 : (−3, −1, −2, −2)
r = 6 : N = 4, n0 = −2, n∞ = −6; T = 13; h1,1 = 35, h2,1 = 11
(C.12)
chain 1 : (−2, −1, −3, −1)
chain 3 : (−2, −1, −3, −1)
chain 4 : (−2, −1, −3, −1)
r = 8 : N = 4, n0 = −2, n∞ = −2; T = 9; h1,1 = 19, h2,1 = 19
(C.13)
chain 1 : (−2, −1, −2)
chain 2 : (−2, −1, −2)
chain 3 : (−2, −1, −2)
chain 4 : (−2, −1, −2)
Note that this last example is a special limit of dP9 with two −2 clusters of type
I0∗ (D̂4 ).
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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